Can I develop a chatbot to carry on a natural conversation with a human using NLP and neural networks?
I would not recommend using neural networks and NLP together to create a system sufficiently capable of conversation/dialogue that it would pass that current crop of Turing-like tests.

Conversations follow certain rules and regularities (which we have only partially discovered so far), and training an ANN with dialogues in order to pick up those regularities is simply not feasible. In conversations you have a memory of what has been mentioned previously, you build up assumptions about the intentions of your dialogue partner, and keep track of the current topic and sub-topics. This is far too complex to be reduced to a machine learning approach.

As a starting point I would suggest looking at ELIZA, developed by Weizenbaum in the mid-1960s. There are plenty of implementations in various programming languages available. Use that as a starting point to extend the capabilities according to topics you want to talk about, and store in memory what the user has said before, trying to refer back to it, etc. This is a lot easier to do with 'symbolic' AI rather than subsymbolic processing.

A lot of current tech companies offer chatbot variants based on machine learning, but they rarely go beyond intent recognition or simple question-answer dialogues. For more sophisticated dialogues they are simply not suitable.

(Disclaimer: I work for a company producing conversational software)